Apparently almost any number can be made into a palindromic number by
reversing the digits and adding and then repeating the steps until you
get a palindromic number. Is there a list that tells which numbers can be
made into palindromic sums and how many steps would be required?

Assume it's proven that any number n>32 can be written in the form n =
m_1 + m_2 + ... + m_k, such that the sum of the inverses of m_i is 1:
1/m_1 + ... + 1/m_k = 1. How can I find an algorithm that prints all
the possible m_i's for a given number n?

The place values in a base 3 number system are powers of 3. Suppose
the digits are 1, 0 and -1. The base 10 number 35 is written as 110-1
in this base 3 system. Write this base 3 notation for the base 10
numbers 1 through 35...

A student knows one solution for Diophantine equations that include a mix of
exponential and algebraic terms. To put reasonable upper bounds in the hunt for
more, Doctor Vogler outlines applications of Baker's Theorem and lattice reduction.

Twenty persons want to buy a $10 ticket each. Ten of them have a $10
note and others have a $20 note. The person at the ticket counter has
no money to start with. What is the probability that the person at the
ticket counter will not have a change problem?

Find all solutions of 2^m = 3^n + 5. The case bothering me is when I
have shown m has to be of form 4k+1 and n of the form 4K+3 for some
k,K and that K has to be even to ensure 3^(4K+3)+5 is congruent to 0
mod 32.

Is it possible to find a formula that takes X (a large number) and
creates another expression which equals X but is shorter in length
than X? For example, take 390,625 which uses six characters. I need
a formula which calculates something like 5^8, which equals 390,625
but is only three characters instead of six.